Generalized Verma modules over some Block algebras

In this paper, a class of generalized Verma modules M(V) over some Block type Lie algebra ℬ(G) are constructed, which are induced from nontrivial simple modules V over a subalgebra of ℬ(G). The irreducibility of M(V) is determined.      

Verma modules over generalized Witt algebras

We constuct and investigate a structure of Verma-like modules over generalized Witt algebras. We also prove Futorny-like theorem for irreducible weight modlues whose dimensions of the weight spaces are uniformly bounded.     

Modules for double affine Lie algebras

Imaginary Verma modules, parabolic imaginary Verma modules,and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated,and several results are generalized from the affine Lie algebras. In particular,imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.     

Certain categories of modules for twisted affine Lie algebras

In this paper, using generating functions, we study two categories ? and ? of modules for twisted affine Lie algebras g^[σ], which were firstly introduced and studied for untwisted affine Lie algebras by H. -S. Li [Math Z, 2004, 248: 635-664]. We classify integrable irreducible g^[σ]-modules in categories ? and ?, where ? is proved to contain the well-known evaluation modules and ? to unify highest weight modules, evaluation modules and their tensor product modules. We determine also the isomorphism classes of those irreducible modules.     

On multiplicities of simple subquotients in generalized Verma modules

We reduce the problem on multiplicities of simple subquotients in an -stratified generalized Verma module to the analogous problem for classical Verma modules.     

Generalized imaginary Verma and Wakimoto modules

We develop a general technique of constructing new irreducible weight modules for any affine Kac-Moody algebra using the parabolic induction, in the case when the Levi factor of a parabolic subalgebra is infinite-dimensional and the central charge is nonzero. Our approach unifies and generalizes all previously known results with imposed restrictions on inducing modules. We also define generalized imaginary Wakimoto modules which provide an explicit realization for generic generalized imaginary Verma modules.     

Coding the principal character formula for affine Kac-Moody lie algebras

In this paper, an algorithm for computing the principal character for affine Lie algebras is discussed and presented. The principal characters discovered using this program are given and/or proven. Results include level 2 and 3 character formulas in and the sole existence of the Rogers-Ramanujan products in , , , , , , .     

Structure of certain Verma modules over affine Lie algebras

Moscow Institute of Electronics Engineering. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 4, pp. 82–83, October–December, 1990.     

Filtrations and completions of certain positive level modules of affine algebras

We define a filtration indexed by the integers on the tensor product of a simple highest weight module and a loop module for a quantum affine algebra. We prove that such a filtration is either trivial or strictly decreasing and give sufficient conditions for this to happen. In the first case we prove that the tensor product is simple and in the second case we prove that the intersection of all the modules in the filtration is zero, thus allowing us to define the completed tensor product. In certain special cases, we identify the subsequent quotients of filtration.     

Classification of Lie algebras of specific type in complexified Clifford algebras

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.     

Categories of Induced Modules and Standardly Stratified Algebras

We construct a generalization of the BGG-category O, whose blocks correspond to standardly stratified algebras. We prove reciprocity formulae in these categories and present two classes of examples.     

Categorification of (induced) cell modules and the rough structure of generalised Verma modules

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (3) (1980) 191-213] in many cases. We also present a general setup of decategorification, precategorification and categorification.     

Free summands of conormal modules and central elements in homotopy Lie algebras of local rings

If is a surjective local homomorphism with kernel , such that and the conormal module has a free summand of rank , then the degree central subspace of the homotopy Lie algebra of has dimension greater than or equal to . This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module.